ROBIN TYPE CONDITIONS ARISING FROM ... - CiteSeerX

Proceedings of Equadiff-11 2005, pp. 157–164 ISBN 978-80-227-2624-5

ROBIN TYPE CONDITIONS ARISING FROM CONCENTRATED POTENTIALS∗ ‡ , AND ANIBAL RODRIGUEZ-BERNAL,§ ´ M. ARRIETA† , ANGELA ´ ´ JOSE JIMENEZ-CASAS,

Abstract. We analyze the limit of the solutions of an elliptic problem with zero flux boundary condition when the potential functions are concentrated in a neighborhood of the boundary and this neighborhood shrinks to the boundary as a parameter goes to zero. We prove that this family of solutions converges in the sup norm to the solution of an elliptic problem with Robin type condition. This Robin type conditions for the limiting problem comes from the concentrated potentials around the boundary of the domain. Key words. Robin flux conditions, elliptic boundary problems, concentrating integrals

1. Introduction. As it is usually accepted that the environmental influence on a reacting media is modelled by a boundary condition in a boundary value problem, it is also common sense to state that boundary conditions of flux type, must also be equivalent to some external action localized in a thin neighborhood of the boundary. In fact elementary application of finite difference schemes to one dimensional boundary value problems, show that this analogy or equivalence must have a sounded mathematical foundation. Therefore in this paper we present a rigurous mathematical analysis of this question for some elliptic (i.e. time independent) linear problems. In particular we show how linear reaction and flux terms on the boundary condition can be obtained as a result of a limiting process from terms distibuted and concentrated near the boundary. 2. Concentrating integrals. Let Ω ⊂ RN be a bounded domain with smooth boundary ∂Ω = Γ and outward normal vector ~n(x) for x ∈ Γ. ¯ We consider the neighborhood of Γ defined as ωε = {x − s~n(x), x ∈ Γ, s ∈ [0, ε)} ⊂ Ω for sufficiently small ε. Then, if we consider Γσ = {x − σ~n(x), x ∈ Γ}, Γ0 = Γ, 0 < ε < ε0 the “parallel” interior boundary, we have ωε = ∪0≤σ q≤

1 p

and s −

N p

≥ − (N q−1) i.e.

p(N −1) N −sp .

Then for ε small enough: R i) The map σ 7→ Γσ v q is continuous. ii) There exists C > 0 independent of ε such that sup kvkLq (Γσ ) ≤ CkvkW s,p (Ω)

(2.1)

σ∈[0,ε) ∗ Partially

supported by Projects BFM2003-03810 and BFM2003-07749-C05-05 / FISI CICYT, Spain. Depto. de Matem´ atica Aplicada, U. Complutense de Madrid, ([email protected]) ‡ Grupo de Din´ amica No Lineal,ETSI de (ICAI) U. Pontificia Comillas de Madrid, ([email protected]) § Depto. de Matem´ atica Aplicada, U. Complutense de Madrid, ([email protected]) †

157

158

J. M. Arrieta, A. Jimen´ ez-Casas and A. Rodr´ıguez-Bernal

iii) Z

ε

Z

|v|q =

ωε

0

Z

|v|q dS dσ.

(2.2)

Γσ

iv) In particular 1 ε

Z ωε

|v|q ≤ ckvkqW s,p (Ω)

(2.3)

and 1 lim ε7→0 ε

Z

Z

q

|v|q

|v| = ωε

Γ

Note that these results assert that for smooth test functions, integrals concentrated close to the boundary converge to integrals on the boundary. With this we can prove then the following result that, in some sense, allows to pass to the limit in concentrating integrals, with nonsmooth integrands. Lemma 2.2. We assume that the functions hε on ωε are such that Z 1 |hε (x)|r dx ≤ C < ∞, for some 1 ≤ r < ∞ ε ωε

(2.4)

for some constant C independent of ε. Then there exists a subsequence, still denoted by ε, and a function h0 ∈ Lr (Γ) (or h0 ∈ M(Γ) if r = 1), such that i) For any continuous function in ωε0 , ϕ, we have Z Z 1 hε (x)ϕ(x) dx = h0 (y)ϕ(y) dσ. (2.5) lim ε7→0 ε ω Γ ε ii) If uε converge to u weakly in W s,p (Ω) with s > 1 ε7→0 ε

Z

1 p

and s −

N p

Z

lim

hε (x)uε dx = ωε

1 ε7→0 ε

(2.6)

h0 (y)u(y) dσ. Γ

iii) If uε 7→ u0 in W s,p (Ω) and ϕε 7→ ϕ0 in W σ,q (Ω) with s > s + σ − Np − Nq > − Nr−1 0 , we have that Z

lim

≥ − Nr−1 0 , then

1 p

and σ >

1 q

and

Z hε (x)uε ϕε (x) dx =

ωε

h0 (y)u0 (y)ϕ(y) dσ.

(2.7)

Γ

iv) In particular if we assume the above the hypothesis, and consider (W σ,q (Ω))0 = 0 0 W −σ,q (Ω) such that W σ,q (Ω) ⊂ Lr (Γ), i.e. σ > 1q and σ − Nq ≥ − Nr−1 0 . Then, 0

1. Lε = 1ε Xωε hε is a bounded family in (W σ,q (Ω))0 := W −σ,q (Ω). 0 2. The operators Bε converge to the operator B0 in L(W s,p (Ω), W −σ,q (Ω)) 1 N 1 N N −1 with s > p and σ > q and s + σ − p − q > − r0 , where < Bε (u), ϕ >=

1 ε

Z

Z hε uϕ

ωε

for u ∈ W s,p (Ω) and ϕ ∈ W σ,q (Ω).

7−→

< B0 (u), ϕ >=

h0 uϕ Γ

(2.8)

159

Equadiff-11. Concentrated potentials

Proof. 1.

If ϕ ∈ W s,p (Ω) from Lemma 2.1, we have that

|Lε (ϕ)| ≤ 2.

If

1 r

+

1 m

1 ε

+

Z

1 n

|hε ||ϕ| ≤ ωε

Z r1 Z 10 0 r 1 1 |hε |r |ϕ|r ≤ CkϕkW s,p (Ω) . ε ωε ε ωε

=1

Z Z r1 Z m1 Z n1 1 1 1 1 r m n hε uϕ ≤ |hε | |u| |ϕ| . ε ε ωε ε ωε ε ωε ωε −1 If r, m, n are such that s0 − pN0 ≥ − Nm with s0 > p10 and σ0 − qN0 ≥ − Nn−1 with σ0 > then from Lemma 2.2 and (2.4) Z 1 hε uϕ ≤ CkukW s0 ,p0 (Ω) kϕkW σ0 ,q0 (Ω) ε ωε

1 q0 ,

0

and Bε from W s0 ,p0 (Ω) into W −σ0 ,q0 (Ω)) is uniformly bounded. Hence, fixed u ∈ W s0 ,p0 (Ω) we have Lemma 2.1 that Z Z 1 < Bε (u), ϕ >= hε uϕ 7→ h0 uϕ =< B0 (u), ϕ > ε ωε Γ uniformly for ϕ in compact sets of W σ0 ,q0 (Ω). Hence if q ≥ q0 with σ > 1q and σ − Nq > σ0 − qN0 then W σ,q (Ω) ⊂ W σ0 ,q0 (Ω) with compact embedding, and then, in particular Bε (u)

7→

0

B0 (u) in W −σ,q (Ω).

Again this implies uniform convergence for u in compact sets of W s0 ,p0 (Ω). Hence if p ≥ p0 with s > p1 and s − Np ≤ s0 − pN0 then W s,p (Ω) ⊂ W s0 ,p0 (Ω) with compact embedding, and then, in particular, we have Bε

7→

B0

0

in L(W s,p (Ω), W −σ,q (Ω))

which gives the result. 3. An elliptic problem with concentrating non homogeneous terms. With the previous results, in this section we study the behavior, for small ε, of the solutions of the elliptic problem    −∆uε + uε = 1 Xωε fε in Ω ε (3.1) ∂u ε   =0 on Γ ∂n with a given fε and where Xωε denote the characteristic function of the set ωε . Hence the effective reaction is concentrated on this set. The goal in this section is to prove that, the family of solutions, uε , converges when the parameter ε goes to zero, to a limit function, which is given by the solution of the homogeneous elliptic problem with nonhomogeneous flux conditions on the boundary: ( −∆u + u = 0 in Ω (3.2) ∂u = f0 on Γ. ∂n

160


Moreover, we give conditions on fε that guarantee that the convergence is uniform ¯ and in C ∞ (Ω). For this we will assume, that fε satisfies in Ω loc Z 1 |fε (x)|r dx ≤ C < ∞ (3.3) ε ωε for some constant C independent of ε and there exists a function, f0 ∈ Lr (Γ) such that for any smooth function ϕ, we have Z Z 1 lim fε (x)ϕ(x) dx = f0 (y)ϕ(y) dσ. (3.4) ε7→0 ε ω Γ ε with ε0 > 0 sufficiently small. With the notations above, we can prove first Theorem 3.1. i) Assume the notations above and the hypotheses (3.3) with r = 1 and (3.4). Then the family uε converges to u in W 1,q (Ω) for every q < NN−1 , in Ls (Ω) for every s < NN−2 and almost everywhere, where u is the unique solution of (3.2). Moreover the family uε also converges in C j (K) for every j and K ⊂⊂ Ω a compact set. ii) Assume in addition that, for r > 1 we have Z 1 |fε (x)|r dx ≤ C < ∞ ε ωε 1

for some constant C independent of ε. Then uε is uniformly bounded in W 1+ r ,r (Ω) and therefore uε with 1 +

1 r

−

N r

>α−

N q .

7→

u in W α,q (Ω)

In particular, if r > N − 1 then uε

7→

¯ u in C β (Ω)

for some β > 0. Proof. i) Let Fε = 1ε Xωε fε which is uniformly bounded in L1 (Ω). Then, there exists λ0 such that if λ ≥ λ0 then the elliptic operator is invertible. Hence, by elliptic regularity we have a uniform bound of solutions, uε , in W 2−δ,1 (Ω) for every δ > 0. Thus, there exists a subsequence which converges weakly to u in W 2−δ,1 (Ω), for every δ > 0. On the other hand, again from Sobolev’s embedding, we have that for any q < NN−1 we can find δ > 0 such that W 2−δ,1 (Ω) ⊂ W 1,q (Ω) with compact inclusion. From this, we can ensure the subsequence converges strongly in W 1,q (Ω) for every q < NN−1 . Moreover, taking into account that, again from Sobolev’s embeddings, for every s < NN−2 we can find q < NN−1 such that W 1,q (Ω) ⊂ Ls (Ω), then the subsequence converge also strongly in Ls (Ω) and thus almost everywhere. Also, the traces of uε converge to the trace of u in 1 W 1− q ,q (Γ). Next, we prove u verifies the homogeneous elliptic problem (3.2). In effect, multiplying the equation from (3.1) by ϕ ∈ C ∞ (Ω) we obtain Z Z 1 (∇uε ∇ϕ + uε )ϕ = fε ϕ. ε ωε Ω


161

Now we assume first ϕ ∈ Cc∞ (Ω) and taking the limit as ε goes to zero, using (3.4), we get Z (3.5) ∇u∇ϕ + uϕ = 0, ∀ϕ ∈ Cc∞ (Ω). Ω ∞ Therefore the limit function satisfies −∆u + u = 0 in Ω. In particular u ∈ Cloc (Ω). ∞ ¯ Now, we consider a test function ϕ ∈ C (Ω) and using again (3.4) together with the convergence of the traces, we get Z Z Z ¯ ∇u∇ϕ + (3.6) uϕ = f0 ϕ, for all ϕ ∈ C ∞ (Ω). Ω

Ω

Γ

Using the equation for u in Ω and integrating by parts, we get now Z Z ∂u ¯ ϕ= f0 ϕ, ∀ϕ ∈ C ∞ (Ω) Γ ∂n Γ where

∂u ∂n

denote the normal derivative defined by Z Z Z ∂u ϕ= ∇u∇ϕ − (−∆u)ϕ Γ ∂n Ω Ω

¯ Therefore u solves (3.2). for every ϕ ∈ C ∞ (Ω). Now, since λ > λ0 , from the uniqueness of solutions for the limit problem (3.2), we have that the whole family uε converges to u. Next, to improve the convergence in Ω, given K ⊂⊂ Ω we consider the auxiliar function vε = uε ϕ where ϕ ∈ Cc∞ (Ω) is a regular function, such that is equal to 1 in K, and 0 in the set Ω \ Ω0 where Ω0 is a domain such that K ⊂ Ω0 ⊂ Ω. Now observe that considering ε small enough such that ωε ⊂ Ω \ Ω0 and thus Fε ϕ = 0 in Ω. Then, vε , satisfies the elliptic problem −∆vε + vε = gε in Ω (3.7) vε = 0 on Γ with gε = −2∇uε ∇ϕ − uε ∆ϕ, which satisfies gε ∈ Lq (Ω). From uniform estimates of the family uε in W 1,q (Ω), with q < NN−1 , we obtain kgε kLq (Ω) ≤ C1 for some constant C1 > 0 independent of ε. Then, from (3.7) there exists C2 > 0 such that kvε kW 2,q (Ω)∩W 1,q (Ω) ≤ C2 . 0 Now, applying a bootstrap argument to vε and taking into account that vε = uε on K we have that, for any j there exists C(j) positive constant independent of ε such that kuε kC j (K) ≤ C(j). Thus, using the Ascoli-Arzela’s theorem, and the uniqueness of u we get the result. ii) From Lemma 2.2 the right hand side in the equation of uε defines a bounded 0 family in the Sobolev’s space (W s,p (Ω))0 = W −s,p (Ω) for s > p1 and s − Np = − Nr−1 0 . Now, from the elliptic regularity, we have the solution of (3.1) satisfies that uε ∈ 0 W 2−s,p (Ω) and is uniformly bounded in this space. 0 Now note that W 2−s,p (Ω) ⊂ W α,q (Ω), with compact embedding, provided 2−s−

N N −1 1 N N =1− =1+ − >α− . p0 r r r q

In particular, if r > N − 1 we have 1 − compact embedding, for some β.

N −1 r

0 ¯ with > 0 and then W 2−s,p (Ω) ⊂ C β (Ω)

162


4. An elliptic problem with concentrating potentials. We assume now that we have concentrating potentials Vε and nonhomogeneous terms fε satisfying Z 1 |Vε (x)|ρ dx ≤ C < ∞ (4.1) ε ωε Z 1 |fε (x)|r dx ≤ C < ∞ (4.2) ε ωε for some constant C independent of ε, and there exists functions f0 ∈ Lr (Γ), and V0 ∈ Lρ (Γ) (or f0 , V0 ∈ M(Γ) if r = ρ = 1), such that for any function ϕ, continuous in ωε0 , we have Z Z 1 lim fε (x)ϕ(x) dx = f0 (y)ϕ(y) dσ. (4.3) ε7→0 ε ω Zε ZΓ 1 lim Vε (x)ϕ(x)dx = V0 (y)ϕ(y)dσ. (4.4) ε7→0 ε ω Γ ε In this section we prove that for such concentrating potentials and nonhomogeneous ¯ where uε satisfies that terms, we get uε 7→ u in C(Ω)  1 1  −∆uε + λuε = ε Xωε Vε uε + ε Xωε fε + g  ∂uε = 0 ∂n and u satisfies

  −∆u + λu = g  ∂u = V u + f 0 0 ∂n

in Ω (4.5) on Γ

in Ω on Γ.

In fact if s, p, σ are as in Lemma 2.2 iii) and if we define the operator Pε ∈ L(W s,p (Ω), W −σ,p (Ω)) as Z 1 < Pε (uε ), ϕ >= Vε uε ϕ, ε ωε then we have Theorem 4.1. Assume Vε satisfies (4.1) with ρ > N − 1. Then there exists some λ0 such that for λ ≥ λ0 the the elliptic operator A0 + λI − Pε in (4.5) is invertible and k(A0 + λI − Pε )−1 kW −σ,p (Ω) ≤

C |λ|

(4.6)

for some constant C independent of ε. Proof. Writing (4.5) as a perturbation of a fixed elliptic operator i.e. A0 uε + λuε = Pε uε + hε from Lemma 2.2 we have Pε 7→ P0 in L(W s,p (Ω), W −σ,p (Ω)) with Z Z 1 < Pε (uε ), ϕ >= Vε uε ϕ 7→ V0 uϕ =< P0 (u), ϕ > ε ωε Γ 0

for every u ∈ W s,p (Ω) and ϕ ∈ W σ,p (Ω).

(4.7)


163

Since ρ > N − 1 we can take s + σ < 2 and we get that A0 + λI − Pε is well defined from W 2−σ,p (Ω) into W −σ,p (Ω). In particular for given g ∈ W −σ,p (Ω) the equation A0 uε + λuε − Pε uε = g can be written as uε = (A0 uε + λI)−1 (g + Pε uε ) = Tλε (uε ). Now the resolvent estimates for A0 imply that the Lipschitz constant of Tλε is bounded C by |λ| , for some constant C independent of ε, and therefore it is a contraction for large enough λ. With this we get Corollary 4.2. Assume that λ ≥ λ0 and in addition that, for r > 1 we have Z 1 |fε (x)|r dx ≤ C < ∞ ε ωε 1

Then uε is uniformly bounded in W 1+ r ,r (Ω) and therefore uε 7→ u in W α,q (Ω) as ε 7→ 0 with 1 +

1 r

−1+

N r

>α−

N q

where u is the unique solution of  in Ω  −∆u + λu = g  ∂u = V0 u + f0 on Γ ∂n In particular, if r > N − 1 then ¯ as ε 7→ 0 uε 7→ u in C β (Ω)

(4.8)

for some β > 0. The proofs above can be extended along the same lines above for more general elliptic problems with smooth coefficients. Theorem 4.3. Under the above notations and the hypotheses for Vε , fε and g, we consider the family uε given by the solutions of the following problems  in Ω −div(a(x)∇uε ) + c(x)uε + λuε = 1ε Xωε Vε uε + 1ε Xωε fε + g     uε = h, on Γ0 , (4.9)   ∂u ε   a(x) + b(x)uε = j on Γ1 ∂n ¯ and b ∈ C 1 (Γ¯1 ) with a(x) ≥ for sufficiently large λ, and smooth coefficients a, c ∈ C 1 (Ω), a0 > 0. ¯ where u is the Then, given h ∈ C(Γ¯0 ) and j ∈ Lr (Γ1 ), we have that uε 7→ u in C(Ω) solution of  −div(a(x)∇u + c(x)u + λu = g     u = h, on Γ0 , (4.10)   ∂u   a(x) + b(x)u = j + V0 u + f0 on Γ1 ∂n The key argument for the proof, relies in considering the scale of extrapolation spaces associated to these operators, as constructed in [1]. In fact it can be shown that this scale does not depend on ε and that the concentrating potentials and nonhomogeneous terms remain uniformly bounded in the corresponding dual norms.

164


5. Singular coefficients. The analysis above fails if the smoothness of the coefficients in (4.9) is removed, since then the scales of spaces in [1] is no longer available. In this case however, by using a different approach we are still able to prove analogous convergence results as above; in particular, we are still able to prove the convergence in ¯ See [2] for details. C(Ω). REFERENCES [1] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), volume 133 of Teubner-Texte Math., Teubner, Stuttgart, 1993. [2] J. M. Arrieta, A. Jim´ enez-Casas and A. Rodr´ıguez-Bernal, Flux terms and Robin boundary condition as limit of reactions and potentials concentrating in the boundary. To appear in Revista Matem´ atica Iberoamericana [3] L. Bocardo, D. Giachet, J. I. D´ıaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear term, J. Differential Equations 106(2) (2000), 215–237. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, SpringerVerlag, ed. New York, 1977. [5] O. Ladyzhenskaya and N. Uralseva, Linear and Quasilinear Elliptic Equations, Academic Press 1968.